Elementary function arithmetic: Difference between revisions

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{{Short description|System of arithmetic in proof theory}}

{{No footnotes|date=November 2017}}

{{redirect|Elementary recursive arithmetic|the computational complexity class|Elementary recursive function}}

In [[proof theory]], a branch of [[mathematical logic]], '''elementary function arithmetic''', also called ('''EFA'''), also called '''elementary arithmetic''' and '''exponential function arithmetic''',<ref>C. Smoryński, "Nonstandard Models and Related Developments" (p. 217). From ''Harvey Friedman's Research on the Foundations of Mathematics'' (1985), Studies in Logic and the Foundations of Mathematics vol. 117.</ref> is the system of arithmetic with the usual elementary properties of 0,&nbsp;1,&nbsp;+,&nbsp;&times;×,&nbsp;''x''<supmath>''x^y''</supmath>, together with [[mathematical induction|induction]] for formulas with [[bounded quantifiersquantifier]]s.

EFA is a very weak logical system, whose [[proof theoretic ordinal]] is ω<supmath>\omega^3</supmath>, but still seems able to prove much of ordinary mathematics that can be stated in the language of [[Peano axioms|first-order arithmetic]].

==Definition==

EFA is a system in first order logic (with equality). Its language contains:

*two constants <math>0</math>, <math>1</math>,

*three binary operations <math>+</math>, &<math>\times;</math>, <math>\textrm{exp}</math>, with <math>\textrm{exp}(''x'',''y'')</math> usually written as ''x''<supmath>''x^y''</supmath>,

*a binary relation symbol <math>< </math> (This is not really necessary as it can be written in terms of the other operations and is sometimes omitted, but is convenient for defining bounded quantifiers).

'''Bounded quantifiers''' are those of the form ∀<math>\forall(x < y)</math> and ∃ <math>\exists(x < y)</math> which are abbreviations for ∀<math>\forall x (x < y)→,,,\rightarrow\ldots</math> and ∃x<math>\exists x(x < y)∧...\land\ldots</math> in the usual way.

The axioms of EFA are

*The axioms of [[Robinson arithmetic]] for <math>0</math>, <math>1</math>, <math>+</math>, &<math>\times;</math>, <math>< </math>

*The axioms for exponentiation: ''x''<supmath>x^0=1</supmath> = 1, ''x''<supmath>''x^{y''+1</sup> }=x^y\times ''x''<sup>''y''</supmath>&times;''x''.

*Induction for formulas all of whose quantifiers are bounded (but which may contain free variables).

==Friedman's grand conjecture<!--'Friedman's grand conjecture' redirects here-->==

[[Harvey Friedman (mathematician)|Harvey Friedman]]'s '''grand conjecture''' implies that many mathematical theorems, such as [[Fermat's lastLast theoremTheorem]], can be proved in very weak systems such as EFA.

The original statement of the conjecture from {{harvtxt|Friedman|1999}} is:

: "Every theorem published in the ''[[Annals of Mathematics]]'' whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in EFA. EFA is the weak fragment of [[Peano Arithmetic]] based on the usual quantifier-free axioms for 0,&nbsp;1,&nbsp;+,&nbsp;&times;×,&nbsp;exp, together with the scheme of [[mathematical induction|induction]] for all formulas in the language all of whose quantifiers are bounded."

While it is easy to construct artificial arithmetical statements that are true but not provable in EFA{{example needed}}, the point of Friedman's conjecture is that natural examples of such statements in mathematics seem to be rare. Some natural examples include [[consistency]] statements from logic, several statements related to [[Ramsey theory]] such as the [[Szemerédi regularity lemma]], and the [[graph minor theorem]], and Tarjan's algorithm for the [[disjoint-set data structure]].

==Related systems==

Several related computational complexity classes have similar properties to EFA:

*One can omit the binary function symbol exp from the language, by taking Robinson arithmetic together with induction for all formulas with bounded quantifiers and an axiom stating roughly that exponentiation is a function defined everywhere. This is similar to EFA and has the same proof theoretic strength, but is more cumbersome to work with.

*There are weak fragments of second-order arithmetic called <math>\mathsf{RCA}_0^*</math> and <math>\mathsf{WKL}_0^*</math> that are conservative over EFA for <math>\Pi_2^0</math> sentences (i.e. any <math>\Pi_2^0</math> sentences proven by <math>\mathsf{RCA}_0^*</math> or <math>\mathsf{WKL}_0^*</math> are already proven by EFA.)<ref>S. G. Simpson, R. L. Smith, "[https://www.sciencedirect.com/science/article/pii/0168007286900746 Factorization of polynomials and <math>\Sigma_1^0</math>-induction]" (1986). [[Annals of Pure and Applied Logic]], vol. 31 (p.305)</ref> In particular, they are conservative for consistency statements. These fragments are sometimes studied in [[reverse mathematics]] {{harv|Simpson|2009}}.

*'''Elementary recursive arithmetic''' ('''ERA''') is a subsystem of [[primitive recursive arithmetic]] (PRA) in which recursion is restricted to [[ELEMENTARY#Definition|bounded sums and products]]. This also has the same <math>\Pi_2^0</math> sentences as EFA, in the sense that whenever EFA proves ∀x∃y ''P''(''x'',''y''), with ''P'' quantifier-free, ERA proves the open formula ''P''(''x'',''T''(''x'')), with ''T'' a term definable in ERA. Like PRA, ERA can be defined in an entirely logic-free{{clarify|date=November 2017}} manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under composition and projection of a ''finite'' number of basis functions, and thus only a finite number of defining equations are needed.

== See also ==

There are weak fragments of second-order arithmetic called RCA{{su|p=*|b=0}} and WKL{{su|p=*|b=0}} that have the same consistency strength as EFA and are conservative over it for Π{{su|p=0|b=2}} sentences, which are sometimes studied in [[reverse mathematics]] {{harv|Simpson|2009}}.

<!-- *[[ELEMENTARY]], a related computational complexity class - already wikilinked in the text as <nowiki>[[ELEMENTARY#Definition|bounded sums and products]]</nowiki> -->

'''Elementary recursive arithmetic''' ('''ERA''') is a subsystem of [[primitive recursive arithmetic]] (PRA) in which recursion is restricted to [[ELEMENTARY#Definition|bounded sums and products]]. This also has the same Π{{su|p=0|b=2}} sentences as EFA, in the sense that whenever EFA proves ∀x∃y P(x,y), with P quantifier-free, ERA proves the open formula P(x,T(x)), with T a term definable in ERA.

* {{annotated link|Elementary function}}

*[[ {{annotated link|Grzegorczyk hierarchy]]}}

*[[ {{annotated link|Reverse mathematics]]}}

* {{annotated link|Ordinal analysis}}

*[[ {{annotated link|Tarski's high school algebra problem]]}}

==References==

Like PRA, ERA can be defined in an entirely logic-free manner, with just the rules of substitution and induction, and defining equations for all elementary recursive functions. Unlike PRA, however, the elementary recursive functions can be characterized by the closure under composition and projection of a ''finite'' number of basis functions, and thus only a finite number of defining equations are needed.

{{reflist}}

== See also ==

* {{Citation | last1=Avigad | first1=Jeremy | title=Number theory and elementary arithmetic | doi=10.1093/philmat/11.3.257 | mr=2006194 | year=2003 | journal=[[Philosophia Mathematica. Philosophy of Mathematics, its Learning, and its Application.]] |series=Series III | issn=0031-8019 | volume=11 | issue=3 | pages=257–284}}

*[[ELEMENTARY]], a related computational complexity class

* {{citation |first=Harvey |last=Friedman |title=grand conjectures |year=1999 |url=http://cs.nyu.edu/pipermail/fom/1999-April/003014.html}}

*[[Grzegorczyk hierarchy]]

* {{Citation | last1=Simpson | first1=Stephen G. |authorlink=Steve Simpson (mathematician)| title=Subsystems of second order arithmetic | url=http://www.math.psu.edu/simpson/sosoa/ | publisher=[[Cambridge University Press]] | edition=2nd | series=Perspectives in Logic | isbn=978-0-521-88439-6 | mr=1723993 | year=2009}}

*[[Reverse mathematics]]

*[[Tarski's high school algebra problem]]

{{Mathematical logic}}

==References==

*{{Citation | last1=Avigad | first1=Jeremy | title=Number theory and elementary arithmetic | doi=10.1093/philmat/11.3.257 | mr=2006194 | year=2003 | journal=Philosophia Mathematica. Philosophy of Mathematics, its Learning, and its Application. Series III | issn=0031-8019 | volume=11 | issue=3 | pages=257–284}}

*{{citation |first=Harvey |last=Friedman |title=grand conjectures |year=1999 |url=http://cs.nyu.edu/pipermail/fom/1999-April/003014.html}}

*{{Citation | last1=Simpson | first1=Stephen G. |authorlink=Steve Simpson (mathematician)| title=Subsystems of second order arithmetic | url=http://www.math.psu.edu/simpson/sosoa/ | publisher=[[Cambridge University Press]] | edition=2nd | series=Perspectives in Logic | isbn=978-0-521-88439-6 | mr=1723993 | year=2009}}

[[Category:Conjectures]]

[[Category:Formal theories of arithmetic]]

[[Category:Proof theory]]

[[Category:Conjectures]]